How to make a circular array with a fixed radius. whose elements gradually scale?
For this, you can use DupliFrames. 

Create a Curve Circle at the desired radius.


In the object to be duplicated, Object tab >'Duplication' panel, choose 'Frames', set the number of frames to the number of duplicates, and uncheck 'Speed', as shown. 
You can also set which axis of the duplicate which follows the path, in
the 'Relations Extras' panel. Here, it's the default.





In the circle's Data tab, check 'Path Animation' with the 'Frames' also set to the number of duplicates, and 'Follow'.if you want the rotation of the duplicates to align to the curve.





CtrlP Parent the duplicate to the circle, with
the 'Follow Path' option. You may have to use AltO on the duplicate to clear its origin to get it to sit on the path.
In a timeline, set up an animation of your duplicated object, and
keyframe its scale at frames 1 and 12, by right-clicking in the Transform > Scale panel of the properties region of the 3D View and selecting 'Set Keyframes' . Here, the scale was 1 on frame
1, and 2 on frame 12.




As indicated in the manual, you can use ShiftCtrlA to make the duplicates real.